Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(10,15)(11,16)(12,13);(1,2,3,4)(5,6)(7,8)(9,10,12,15,14,16,13,11);(6,8)(9,11,13,16,14,15,12,10);(6,8);(10,16)(11,15)Polycyclic presentation:
2^9|b^2=f,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72226283, #72228311, #72228332, … |
| maximal normal subgroup of | #72226283, #72228311, #120×AGL(1,ℤ/8), … |
| maximal quotient of | #72214731, #72214773, #72214806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350440);